Problem 73 - Elementary Algebra

Question

For the following problems, convert the given rational expressions to rational expressions having the same denominators. $$ \frac{8}{z}, \frac{3}{4 z^{3}} $$

Step-by-Step Solution

Answer

Question: Convert the given rational expressions to expressions with the same denominator and write the final rational expressions: $$ \frac{8}{z}, \frac{3}{4 z^{3}} $$ Answer: The given rational expressions with the same denominator are: $$ \frac{32z^2}{4z^3}, \frac{3}{4z^3} $$

Step 1: 1. Identify the given rational expressions

The given rational expressions are: $$ \frac{8}{z}, \frac{3}{4 z^{3}} $$

Step 2: 2. Find the least common multiple (LCM) of the denominators

In order to find the LCM of the denominators, we will first need to factorize the denominators: $$ z = z \\ 4 z^{3} = 2^2 \cdot z^3 $$ Now, we can find the LCM by taking the highest powers of each factor: $$ \text{LCM}(z, 4z^3) = 2^2 \cdot z^3 = 4z^3 $$

Step 3: 3. Rewrite each rational expression with the common denominator

Now that we have the LCM, we can rewrite each rational expression with the common denominator \(4z^3\). We do this by multiplying both the numerator and the denominator of each expression by the necessary factors to get the common denominator: $$ \frac{8}{z} = \frac{8}{z} \cdot \frac{4z^2}{4z^2} = \frac{32z^2}{4z^3} \\ \frac{3}{4 z^{3}} = \frac{3}{4 z^{3}} \cdot \frac{1}{1} = \frac{3}{4z^3} $$

Step 4: 4. Write the final rational expressions

The given rational expressions, converted to have the same denominator, are: $$ \frac{32z^2}{4z^3}, \frac{3}{4z^3} $$

Key Concepts

Common Denominators

When dealing with rational expressions, having a common denominator is essential for comparing, adding, or subtracting them. Simply put, a common denominator is a shared multiple of the denominators of two or more fractions. For example, in the expressions \( \frac{8}{z} \) and \( \frac{3}{4z^3} \), these fractions share different denominators, \( z \) and \( 4z^3 \).
To perform operations on these rational expressions effectively, each must be expressed in terms of a common denominator.

• Identifying a common denominator facilitates straightforward addition or subtraction of fractions.
• The process involves rewriting the fractions so that their denominators are the same.

This rewriting step is crucial for transforming fractions to be on the same playing field, allowing you to handle them with ease.

Least Common Multiple (LCM)

To find a common denominator, we first need to determine the Least Common Multiple (LCM) of the denominators involved. The LCM is the smallest multiple that is evenly divisible by each denominator. Here's how the process unfolds, using our previous example:\( z \) and \( 4z^3 \).

First, we factor the denominators:

• \( z = z \)
• \( 4z^3 = 2^2 \cdot z^3 \)

By identifying all the factors, the LCM is formed by choosing the highest power of each factor present in any of the denominators.

• In this case: \( \text{LCM}(z, 4z^3) = 2^2 \cdot z^3 = 4z^3 \)

The LCM ensures that all rational expressions are expressed on a common scale.

Simplifying Expressions

Once we have the least common denominator, we can simplify the given rational expressions. This involves adjusting each fraction so that it has the same denominator as the LCM.

For the expression \( \frac{8}{z} \), it needs to be rewritten as follows:

• Multiply the numerator and denominator by \( 4z^2 \) to get: \( \frac{8}{z} = \frac{32z^2}{4z^3} \)

For the expression \( \frac{3}{4z^3} \), since it already has the denominator \( 4z^3 \), it becomes:

• \( \frac{3}{4z^3} = \frac{3}{4z^3} \)

Essentially, simplifying in this context means making sure that both expressions have a common denominator, allowing for easy manipulation, comparison, or further operations like adding or subtracting. With expressions harmonized in their denominators, math becomes much simpler and more intuitive.